Question:
Mark the correct alternative in the following:
If the function $f(x)=2 x^{2}-k x+5$ is increasing on $[1,2]$, then $k$ lies in the interval.
A. $(-\infty, 4)$
B. $(4, \infty)$
C. $(-\infty, 8)$
D. $(8, \infty)$
Solution:
Formula:- The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$
$f(x)=2 x^{2}-k x+5$
$\mathrm{d}\left(\frac{\mathrm{f}(\mathrm{x})}{\mathrm{dx}}\right)=4 \mathrm{x}-\mathrm{k}=\mathrm{f}^{\prime}(\mathrm{x})$
$f^{\prime}(x)>0$
$\Rightarrow 4 x-k>0$
$\Rightarrow K<4 x$
For $x=1$
$\Rightarrow K<4$
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